Probability STEADY - STATE GI / GI / n QUEUE IN THE HALFIN - WHITT REGIME

We consider the FCFS GI/GI/n queue in the so-called HalfinWhitt heavy traffic regime. We prove that under minor technical conditions the associated sequence of steady-state queue length distributions, normalized by n 1 2 , is tight. We derive an upper bound on the large deviation exponent of the limiting steady-state queue length matching that conjectured by Gamarnik and Momcilovic in [16]. We also prove a matching lower bound when the arrival process is Poisson. Our main proof technique is the derivation of new and simple bounds for the FCFS GI/GI/n queue. Our bounds are of a structural nature, hold for all n and all times t ≥ 0, and have intuitive closed-form representations as the suprema of certain natural processes which converge weakly to Gaussian processes. We further illustrate the utility of this methodology by deriving the first non-trivial bounds for the weak limit process studied in [37].